{
  "abstract": "We study properties of compact Besicovitch sets and other Kakeya-type sets which are typical in the sense that the exceptions form a meager set in the Baire category sense in a suitable metric space. The main motivation comes from the known fact that typical compact Besicovitch sets in $\\mathbb{R}^n$ have minimal Hausdorff dimension among all Besicovitch sets of $\\mathbb{R}^n$. Thus, in order to estimate the smallest possible Hausdorff dimension of an arbitrary Besicovitch set, it is enough to estimate the Hausdorff dimension of a typical (compact) Besicovitch set, or analogously of a Besicovitch set satisfying our list of typical properties. Among other results, we prove that a typical Besicovitch set $K$ in $\\mathbb{R}^n$ has the following properties: it is not sticky, any line intersects $K$ in a set of ($1$-dimensional) measure zero or in a unit line segment and a set of measure zero, it has no more than $n$ different unit line segments in any direction, and for each $1\\le j\\le n$ the Hausdorff dimension of the set of those directions in which $K$ has exactly $j$ unit line segments is $n-j$. For compact metric spaces $X$ and $Y$, let $\\mathcal{C}(X,Y)$ denote the set of compact subsets of $X\\times Y$ whose projection onto the first coordinate is $X$, equipped with the Hausdorff metric. Kakeya-type compact sets can be coded by elements of such spaces, allowing typical properties to be studied through comeager subsets of $\\mathcal C(X,Y)$. Among other results, we prove that, under suitable conditions on $X$ and $Y$, a typical $A\\in \\mathcal{C}(X,Y)$ is totally disconnected and purely unrectifiable, its projection to the second coordinate has Hausdorff dimension zero, and the dimension identities $\\dim_H(A)=\\dim_H(X)$ and $\\dim_P(A)=\\dim_P(X)+\\dim Y$ hold.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37419",
  "author_total": 3,
  "authors": [
    {
      "name": "Josef Bell"
    },
    {
      "name": "Ezekiel Fleissner-Kates"
    },
    {
      "name": "Tamás Keleti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fw2tsarsnz89s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37419",
  "fulltext_url": null,
  "id": "fw2tsarsnz89s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37419",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7224080174",
  "pdf_url": "https://arxiv.org/pdf/2610.37419",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fw2tsarsnz89s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37419",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37419"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Typical Kakeya-type sets and their typical code sets",
  "title_text": "Typical Kakeya-type sets and their typical code sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/fw2tsarsnz89s",
  "venue": "arXiv",
  "versions": []
}